The function $f(x)=1/(1+x^2)$ belongs to this class but not to the Schwarz space.

Any $C^\infty$ function with bounded support belongs to the Schwarz space but not to this class.

The function $f(x)=1/(1+x^2)$ belongs to this class but not to the Schwartz space of test functions.

Any $C^\infty$ function with bounded support belongs to the Schwartz space but not to this class.